A Review for an Isotropic Landau Model

Abstract

We consider the equationThis model has attracted some attention in the recents years and several results are available in the literature. We review recent results on existence and smoothness of solutions and explain the open problems.

On the gradient flow structure of the isotropic Landau equation

“…which is called the isotropic Landau equation since A[ρ] is replaced by Lρ and has been studied by Gualdani and her collaborators in papers [10,12,11]. We recall that the inverse fractional Laplacian operator (−∆) −s is a Riesz potential of order 2s and can be expressed as…”

“…Later, the radial symmetry requirement was relaxed to even functions in [12]. Although the isotropic Landau equation (1.3) is structurally similar to the classical Landau-Coulomb equation, in analysis it is very different in the sense that its second moment increases in time as shown in [11]. Because of that, the dissipation computation…”

“…Let us summarize some conditional regularity results from [12,11], which will be utilized to analyze a distance based on the nonlocal mobility to be introduced below. By assuming that the initial data ρ 0 is even, |E(ρ 0 )| < +∞ and ||ρ 0 || L 1 = 1 by normalization, the following timedependent dissipation-Fisher information relation holds…”

“…The conditional regularity estimates in [12,11] are built on assuming the following ε−Poincaré inequality…”

“…The goal of this short note is to identify the gradient flow structure of a modified version of classical Landau-Coulomb equations, the isotropic Landau equation, which has drawn interests within the kinetic community recently [14,9,10,12,11]. Given this gradient flow structure, we are able to characterize some basic geometric properties and calculate a time-dependent convergence rate for the entropy dissipation.…”